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1、SOLUTIONS/HINTSTOTHEEXERCISESFROMCOMPLEXANALYSISBYSTEINANDSHAKARCHIROBERTC.RHOADESAbstract.ThiscontainsthesolutionsorhintstomanyoftheexercisesfromtheComplexAnalysisbookbyEliasSteinandRamiShakarchi.IworkedtheseproblemsduringtheSpringof2006whileIwastakingaComplexAnalysiscoursetaughtbyAndreasSeegerat
2、theUniversityofWisconsin-Madison.Iamgratefultohimforhiswonderfullecturesandhelpfulconversationsaboutsomeoftheproblemsdiscussedbelow.Contents1.Chapter1.PreliminariestoComplexAnalysis22.Chapter2.Cauchy’sTheoremandItsApplications83.Chapter3.MeromorphicFunctionsandtheLogarithm94.Chapter4.TheFourierTra
3、nsform105.Chapter5:EntireFunctions116.Chapter6.TheGammaandZetaFunctions137.Chapter7:TheZetaFunctionandPrimeNumberTheorem178.Chapter8:ConformalMappings209.Chapter9:AnIntroductiontoEllipticFunctions2310.Chapter10:ApplicationsofThetaFunctions25Date:September5,2006.TheauthoristhankfulforanNSFgraduater
4、esearchfellowshipandaNationalPhysicalScienceConsortiumgraduatefellowshipsupportedbytheNSA.12ROBERTC.RHOADES1.Chapter1.PreliminariestoComplexAnalysisExercise1.Describegeometricallythesetsofpointszinthecomplexplanedefinedbythefollowingrelations:(1)
5、z−z1
6、=
7、z−z2
8、wherez1,z2∈C.(2)1/z=z.(3)Re(z)=3.(4)Re(z
9、)>c,(resp.,≥c)wherec∈R.(5)Re(az+b)>0wherea,b∈C.(6)
10、z
11、=Re(z)+1.(7)Im(z)=cwithc∈R.Solution1.(1)Itisthelineinthecomplexplaneconsistingofallpointsthatareanequaldistancefrombothz1andz2.Equivalentlytheperpendicularbisectorofthesegmentbetweenz1andz2inthecomplexplane.(2)Itistheunitcircle.(3)Itisthelinewhe
12、reallthenumbersonthelinehaverealpartequalto3.(4)Inthefirstcaseitistheopenhalfplanewithallnumberswithrealpartgreaterthanc.Inthesecondcaseitistheclosedhalfplanewiththesamecondition.(5)(6)Calculate
13、z
14、2=x2+y2=(x+1)2=x2+2x+1.Soweareleftwithy2=2x+1.Thusthecomplexnumbersdefinedbythisrelationisaparabolaopen
15、ingtothe“right”.(7)Thisisaline.Exercise2.Leth·,·idenotetheusualinnerproductinR2.Inotherwords,ifZ=(z,y)and11W=(x2,y2),thenhZ,Wi=x1x2+y1y2.Similarly,wemaydefineaHermitianinnerproduct(·,·)inCby(z,w)=zw.ThetermHermiti
